In number theory, Dirichlet characters are certain arithmetic functions that capture some important properties of the cyclic group. Dirichlet characters are used to define Dirichlet L-functions, meromorphic functions which have a variety of interesting analytic properties. An important special case of an L-function is the Riemann zeta function. Just as the Riemann zeta function is conjectured to obey the Riemann hypothesis, so the L-functions are conjectured to obey the generalized Riemann hypothesis. They are named in honour of Johann Peter Gustav Lejeune Dirichlet.

Axiomatic definition

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A Dirichlet character is a function χ from the integers to the complex numbers which has the following properties:

1. There exists a positive integer k such that χ(n) = χ(n + k) for all n.
2. χ(n) = 0 for every n with gcd(n,k) > 1.
3. χ(mn) = χ(m)χ(n) for all integers m and n.
4. χ(1) = 1.

Condition 1. shows says that the character is periodic with period k. Condition 3. says that characters are completely multiplicative. The unique character of period 1 is called the trivial character. Note that any character vanishes at 0 except that trivial one, which is 1 on all integers.

Construction via residue classes

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The last two properties show that every Dirichlet character χ is completely multiplicative. One can show that χ(n) is a φ(k)th root of unity whenever n and k are coprime, and where φ(k) is the totient function. Dirichlet characters may be viewed in terms of the character group of the unit group of the ring Z/kZ, as given below.

Residue classes

Given an integer k, one defines the residue class of an integer n as the set of all integers congruent to n modulo k: $\backslash hat\{n\}=\backslash \{m\; |\; m\; \backslash equiv\; n\; \backslash mod\; k\; \backslash \}.$ That is, the residue class $\backslash hat\{n\}$ is the coset of n in the quotient ring Z/kZ.

The set of units modulo k forms an abelian group of order $\backslash phi(k)$, where group multiplication is given by $\backslash hat\{mn\}=\backslash hat\{m\}\backslash hat\{n\}$ and $\backslash phi$ denoted Euler's phi function. The identity in this group is the residue class $\backslash hat\{1\}$ and the inverse of $\backslash hat\{m\}$ is the residue class $\backslash hat\{n\}$ where $mn=1\; \backslash mod\; k$. For example, for k=6, the set of units is $\backslash \{\backslash hat\{1\},\; \backslash hat\{5\}\backslash \}$ because 0, 2, 3, and 4 are not coprime to 6.

Dirichlet characters

A Dirichlet character modulo k is a group homomorphism $\backslash chi$ from the unit group modulo k to the non-zero complex numbers (necessarily with values that are roots of unity since the units modulo k form a finite group). We can lift $\backslash chi$ to a completely multiplicative function on integers relatively prime to k and then to all integers by extending the function to be 0 on integers having a non-trivial factor in common with k. The principal character $\backslash chi\_1$ modulo k has the properties

$\backslash chi\_1(n)=1$ if gcd(n, k) = 1 and

$\backslash chi\_1(n)=0$ if gcd(n, k) > 1.

When k is 1, the principal character modulo k is equal to 1 at all integers. For k greater than 1, the principal character modulo k vanishes at integers having a non-trivial common factor with k and is 1 at other integers.

A few character tables

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The tables below help illustrate the nature of a Dirichlet character. They present all of the characters from modulus 1 to modulus 7.

Modulus 1

There is one (1 = φ(1)) character modulo 1:

{| cellspacing="1" cellpadding="3" style="text-align:right;" χ \ n   1 $\backslash chi\_1(n)$ 1

Modulus 2

There is exacly one (1=φ(2)) character to the modulus 2:

{| cellspacing="1" cellpadding="3" style="text-align:right;" χ \ n   1 2 $\backslash chi\_1(n)$ 1 0

Modulus 3

There are $\backslash phi(3)=2$ characters modulo 3:

{| cellspacing="1" cellpadding="3" style="text-align:right;" χ \ n   1 2 3 $\backslash chi\_1(n)$ 1 1 0 $\backslash chi\_2(n)$ 1 −1 0

Modulus 4

There are $\backslash phi(4)=2$ characters modulo 4:

{| cellspacing="1" cellpadding="3" style="text-align:right;" χ \ n   1 2 3 4 $\backslash chi\_1(n)$ 1 0 1 0 $\backslash chi\_2(n)$ 1 0 −1 0

The Dirichlet L-series (defined below) for $\backslash chi\_1(n)$ is

$L(\backslash chi\_1,\; s)=\; (1-2^\{-s\})\backslash zeta(s)\backslash ,$

where $\backslash zeta(s)$ is the Riemann zeta-function. The L-series for $\backslash chi\_2(n)$ is the Dirichlet beta-function

$L(\backslash chi\_2,\; s)=\backslash beta(s).\backslash ,$

Modulus 5

There are $\backslash phi(5)=4$ characters modulo 5. In the tables, i is a square root of $-1$.

{| cellspacing="1" cellpadding="3" style="text-align:right;" χ \ n   1 2 3 4 5 $\backslash chi\_1(n)$ 1 1 1 1 0 $\backslash chi\_2(n)$ 1 −1 −1 1 0 $\backslash chi\_3(n)$ 1 i −i −1 0 $\backslash chi\_4(n)$ 1 −i i −1 0

Modulus 6

There are $\backslash phi(6)=2$ characters modulo 6:

{| cellspacing="1" cellpadding="3" style="text-align:right;" χ \ n   1 2 3 4 5 6 $\backslash chi\_1(n)$ 1 0 0 0 1 0 $\backslash chi\_2(n)$ 1 0 0 0 −1 0

Modulus 7

There are $\backslash phi(7)=6$ characters modulo 7. In the table below, $\backslash omega\; =\; \backslash exp(\; \backslash pi\; i\; /3).$

{| cellspacing="1" cellpadding="3" style="text-align:right;" χ \ n     1     2     3     4     5     6     7   $\backslash chi\_1(n)$ 1 1 1 1 1 1 0 $\backslash chi\_2(n)$ 1 1 −1 1 −1 −1 0 $\backslash chi\_3(n)$ 1 ω² −ω ω −ω² −1 0 $\backslash chi\_4(n)$ 1 ω² ω ω ω² 1 0 $\backslash chi\_5(n)$ 1 ω ω² ω² ω 1 0 $\backslash chi\_6(n)$ 1 ω -ω² ω² −ω −1 0

Examples

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If p is a prime number, then the function

$\backslash chi(n)\; =\; \backslash left(\backslash frac\{n\}\{p\}\backslash right),\backslash$

where $\backslash left(\backslash frac\{n\}\{p\}\backslash right)$ is the Legendre symbol, is a Dirichlet character modulo p.

Dirichlet L-series

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If χ is a Dirichlet character, one defines its Dirichlet L-series by

$L(\backslash chi,s)\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{\backslash chi(n)\}\{n^s\}$

where s is a complex number with real part > 1. By analytic continuation, this function can be extended to a meromorphic function on the whole complex plane. Dirichlet L-series are generalizations of the Riemann zeta-function and appear prominently in the generalized Riemann hypothesis.

Relation to the Hurwitz zeta-function

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The L-functions may be written as a linear combination of the Hurwitz zeta-function at rational values. Fixing an integer k ≥ 1, the Dirichlet L-functions for characters modulo k are linear combinations, with constant coefficients, of the ζ(s,q) where q = m/k and m = 1, 2, ..., k. This means that the Hurwitz zeta-function for rational q has analytic properties that are closely related to the Dirichlet L-functions. Specifically, let $\backslash chi$ be a character modulo k. Then we can write its Dirichlet L-function as

$L(\backslash chi,\; s)\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\; \{\backslash chi(n)\}\{n^s\}\; =$

\frac {1}{k^s} \sum_{m=1}^k \chi(m)\; \zeta \left(s,\frac{m}{k}\right) .

In particular, the Dirichlet L-function of the trivial character modulo 1 yields the Riemann zeta-function:

$\backslash zeta(s)\; =\; \backslash frac\; \{1\}\{k^s\}\; \backslash sum\_\{m=1\}^k\; \backslash zeta\; \backslash left(s,\backslash frac\{m\}\{k\}\backslash right)$

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History

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Dirichlet characters and their L-series were introduced by Johann Peter Gustav Lejeune Dirichlet, in 1831, in order to prove Dirichlet's theorem about the infinitude of primes in arithmetic progressions. He only studied them for real s and especially as s tends to 1. The extension of these functions to complex s in the whole complex plane was obtained by Bernhard Riemann in 1859.

See also

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• Character sum
• Gaussian sum
• Primitive root modulo n
• Selberg class

References

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• Tom M. Apostol Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0-387-90163-9 See chapter 6.

Zeta and L-functions

Caractère de Dirichlet | Dirichlet-karakter